I need help in creating a new profit function in a scenario where price discrimi

Question

I need help in creating a new profit function in a scenario where price discrimination is not allowed. Below is the full detail including the answer. Please keep in mind I am not asking you to solve the entire problem - I need you to break down the basic algebra steps I am missing in creating the new profit function. Thanks in advance.

American Export-Import Shipping company operates a general cargo carrier service between New York and several Western European ports. It hauls two major categories of freight: manufactured items and semi manufactured raw materials.The demand functions for these two classes of goods are:
P1 = 100 - 2Q1 and P2 = 80 - Q2

Q is tons of freight moved.

TC = 20 + 4(Q1 + Q2)

If American is required by law to charge the same per-ton rate to all users, calculate the new profit maximizing level of price and output. What are the profits in this situation?

Let me show you what I have:
P1 = P2 = P
100 - 2Q1 = 80 - Q2
Q2 = -20 + 2Q1
p = -1940 + 328Q1 - 6Q12 *****Here is where I am lost. How did they come up with this?

I can solve from here on - just need help in the how (basic detail please) for the profit function above.

dp/dQ1 = 328 - 12Q1 = 0
Q*1 = 27.3; Q*2 = -20 + 2(27.3) = 34.6
Q* = 27.3 + 34.6 = 61.9
P* = 100 - 2(27.3) = $45.4
p * = (45.4)(61.9) - 20 - 4(61.9)
= $2542.66

Thank you for your time - this one section has been frustrating for me.

Explanation / Answer

P1 = P2 = P 100 - 2Q1 = 80 - Q2 Q2 = -20 + 2Q1 -------------------------------- Q is tons of freight moved. TC = 20 + 4(Q1 + Q2) TC = 20 + 4(Q1 + (-20+2Q1) TC = 20 + 4Q1 -80 + 8Q4 You should be able to plug in the values of Q1 and Q2 and then when you solve for a system of equations end up with: p = -1940 + 328Q1 - 6Q12

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